Part 1 - Pre-Algebra Summary Page 1 22Part 1 - Pre-Algebra Summary Page of 22 + – + + = + > • 2....

Part 1 - Pre-Algebra Summary of 22

1/19/12

Copyright © 2006-2012 Sally C. Zimmermann. All rights reserved.

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Table of Contents

1. Numbers .................................................. 2

1.1. NAMES FOR NUMBERS ........................................................................................................................................... 2

1.2. PLACE VALUES ...................................................................................................................................................... 3

1.3. INEQUALITIES ........................................................................................................................................................ 4

1.4. ROUNDING ............................................................................................................................................................. 4

1.5. DIVISIBILITY TESTS ............................................................................................................................................... 5

1.6. PROPERTIES OF REAL NUMBERS ............................................................................................................................ 5

1.7. PROPERTIES OF EQUALITY ..................................................................................................................................... 5

1.8. ORDER OF OPERATIONS ......................................................................................................................................... 6

2. Real Numbers .......................................... 7

2.1. OPERATIONS .......................................................................................................................................................... 7

3. Fractions ................................................. 8

3.1. DEFINITIONS .......................................................................................................................................................... 8

3.2. LEAST COMMON DENOMINATORS.......................................................................................................................... 9

3.3. OPERATIONS ........................................................................................................................................................ 10

4. Decimals ................................................ 11

4.1. OPERATIONS ........................................................................................................................................................ 11

5. Conversions ........................................... 12

5.1. PERCENT TO DECIMAL TO FRACTIONS ................................................................................................................. 12

6. Algebra .................................................. 13

6.1. DEFINITIONS ........................................................................................................................................................ 13

6.2. OPERATIONS OF ALGEBRA ................................................................................................................................... 14

6.3. SOLVING EQUATIONS WITH 1 VARIABLE.............................................................................................................. 15

6.4. SOLVING FOR A SPECIFIED VARIABLE .................................................................................................................. 16

6.5. EXPRESSIONS VS. EQUATIONS .............................................................................................................................. 17

7. Word Problems ...................................... 18

7.1. VOCABULARY ...................................................................................................................................................... 18

7.2. FORMULAS ........................................................................................................................................................... 19

7.3. STEPS FOR SOLVING ............................................................................................................................................. 20

7.4. RATIOS & PROPORTIONS ...................................................................................................................................... 21

7.5. GEOMETRY .......................................................................................................................................................... 22


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1. Numbers 1.1. Names for Numbers

Real Numbers

(points on a number line)

Irrationals (can’t be written

as a quotient of integers) 3.1415927....

2 1.41421356...

π =

=

Rationals (can be written as a

quotient of integers)

34

241

33%100

7.710

= ±

=

=

Integers

…,–3, –2, –1, 0, 1, 2, 3,…

Whole Numbers

0, 1, 2, 3, …

Natural Numbers

1, 2, 3…


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1.2. Place Values

HU

ND

RE

D-

MIL

LIO

NS

TE

N-M

ILL

ION

S

MIL

LIO

NS

HU

ND

RE

D-

TH

OU

SA

ND

S

TE

N-

TH

OU

SA

ND

S

TH

OU

SA

ND

S

HU

ND

RE

DS

TE

NS

ON

ES

“AN

D”

TE

NT

HS

HU

ND

RE

DT

HS

TH

OU

SA

ND

TH

S

TE

N-

TH

OU

SA

ND

TH

S

9 8 7 6 5 4 3 2 1 . 1 2 3 4

10

0,0

00

,00

0

10

.00

0.0

00

1,0

00

,00

0

10

0,0

00

10

,00

0

10

00

10

0

10

1

Dec

imal

Po

int 1

10

1

100

1

1000

1

10,000

� Place values – go up by powers of 10. 234 can be expressed as (2 100) (3 10) (4 1)• + • + •

� Commas – In numbers with more than 4 digits, commas separate off each group of 3 digits,

starting from the right. These groups are read off together.

� Reading numbers – 123,406,009.023 = “One hundred twenty-three million, four hundred six

thousand, nine and twenty-three thousandths”


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1.3. Inequalities < Less than > Greater than ≤ Less than or equal to ≥ Greater than or equal to = Equal ≠ Not equal to

� Mouth “eats” larger number

� Represent numbers on a number line. The

number to the right is always greater

–7 –6 0 6 7

� Convert numbers so everything is “the same” –

e.g. all decimals, all fractions with common

denominators, etc.

> 7 > 6 > 6 < 7 > –7 <– 6 > –6 > –7

> –5/3 ? –.01

–.67 < –.01

> –4/2 ? –10/4

–8/4 > –10/4

1.4. Rounding 1. Locate the digit to the right of the given place value

2. If the digit is ≥ 5, add 1 to the digit in the given place value

3. If the digit is < 5, the digit in the given place value remains

4. Zero/drop remaining digits

Round to the nearest hundre

1200ds:

12 012 5.12 1 00

25 3

==

>

>

Round to the nearest hundredths

25.13:

25.12 425

53.12 0 25.12

==

>

>


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1.5. Divisibility Tests 2 If last digit is 0,2,4,6, or 8 22, 30, 50, 68, 1024

3 If sum of digits is divisible by 3 123 is divisible by 3 since 1 + 2 + 3 = 6 (and 6 is

divisible by 3)

4 If number created by the last 2 digits is

divisible by 4

864 is divisible by 4 since 64 is divisible by 4

5 If last digit is 0 or 5 5, 10, 15, 20, 25, 30, 35, 2335

6 If divisible by 2 & 3 522 is divisible by 6 since it is divisible by 2 & 3

9 If sum of digits is divisible by 9 621 is divisible by 9 since 6 + 2 + 1 = 9 (and 9 is

divisible by 9)

10 If last digit is 0 10, 20, 30, 40, 50, 5550

1.6. Properties of Real Numbers

For Addition For Subtraction For Multiplication

For Division

Commutative a + b = b + a a – b ≠ b – a ab = ba a/b ≠ b/a

Associative (a+b)+c = a+(b+c) (a–b)–c ≠ a–(b–c) (ab)c = a(bc) (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)

Identity 0+a = a & a+0 = a a – 0 = a a• 1=a & 1 • a=a a ÷ 1 = a

Inverse a + (–a) = 0 &

(–a) + a = 0 a – a = 0

1/a• a = 1 &

a• 1/a =1 if a ≠ 0 a ÷ a = 1 if a ≠ 0

Distributive Property

a(b + c) = ab + ac a(b – c) = ab – ac –a(b + c) = –ab – ac –a(b – c) = –ab + ac

1.7. Properties of Equality

Addition Property of Equality If a = b then a + c = b + c

Multiplication Property of Equality If a = b then ac = bc

Multiplication Property of 0 0 • a = 0 and a • 0 = 0


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1.8. Order of Operations

1111 SIMPLIFY ENCLOSURE SYMBOLS:

Absolute value , parentheses ( ), or brackets [ ]

� If multiple enclosure symbols, do innermost 1st

� If fraction, pretend it has ( ) around its

numerator and ( ) around its denominator

[ ]3 12

2 3( )

12

2

2

2 1

3 2

3(3)

6 12

2( 9)

18

( )

−

+

×

−

+

+

−

−

+=

=

= −

>

18

1= −

2222 CALCULATE EXPONENTS

(Left to Right)

2 2

2

2 2

2

2

2

Evaluate for 5 (means the base is 5, not 5!It's very helpful to put the valueof the variable in parentheses)

( 5) ( 5)( 5)

( 5) ( 5)( 5) (25)

5 5

5 5

25

25

25

25

x

x x = −−

= − = − − =

− − = − − − = − =

= =

− = =

−

>

>

>

>

3333 PERFORM MULTIPLICATION & DIVISION

(Left to Right)

2 10 30 105 2

5 20 3 2

− + •

= −

•

+

÷

•

>

4444 PERFORM ADDITION & SUBTRACTION

(Left to Right)

5 20 6

5 ( 20) 6

15 6

9

= − +

= + − +

= +

= −

−

5555 SIMPLIFY FRACTIONS 5

2

3

5

1

24 6 2

36 6 2

30 3

9

10 1 OR

2

3

10

33

3 33

=

• •= =

• •

•=

•/=

>

>

>


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2. Real Numbers

2.1. Operations Absolute Value

x

� The DISTANCE (which is always

positive) of a number from zero on

the number line

2

2

2

2

00

=

=−

=

>

>

>

2

2

2

2 (2 2

2

)

2

4

=

=

−

−

−

= • −

−

− =−

−

−

>

>

>

Addition +

� If the signs of the numbers are the

SAME, ADD absolute values

� If the signs of the numbers are

DIFFERENT, SUBTRACT absolute

values

� The answer has the sign of the

number with the largest absolute

value

Subtraction _

� Change subtraction to ADDITION OF

THE OPPOSITE NUMBER

� ADD numbers AS ABOVE

1

differen

3 3

minuend

2

2 ( 2)

subtrahend

( 3) 3

2 ( 2)

( 2)

2

3

ce

3 3

3 2

− = =

− − = −− − = −− −

+ −

− ++ −

− += −

>

>

>

>

Multiplication •

� MULTIPLY the numbers

� DETERMINE THE SIGN OF THE

ANSWER

� If the number of negative signs

is EVEN, the answer is POSITIVE

� If the number of negative signs

is ODD, the answer is NEGATIVE

6

produc

3 2 3 2 (3)(2)

factor factor

( 3)( 2)

3 2 3 ( 2) (3)( 2)

(

t

6

6

123)( 2)( 2

)

• = × = =

=

• = • =

− −

− −

−

−

−

=

=

−

− −

>

>

>

>

Division ÷ (Divisors

≠ zero)

� DIVIDE the numbers

� DETERMINE THE SIGN of the answer

by using the MULTIPLICATION SIGN

RULES AS ABOVE

( 2)

divisor quotien

( 12)

dividend

( 12)

/

1

t

6

2 2 6

2

6÷ =

=

−

−

−

−

− =

−

>

>

>

Exponential Notation

� A exponent is a shorthand way to

show how many times a number

(the base) is multiplied by itself

� An exponent applies only to the

base

� Any number to the zero power is 1

0

1

2

1333 3 9

3

3

=== • =

>

2

2

2

2

2 2( )

(base is

5 5 52 5 2 2 5 52 2 2 2 2 2

2)( ) (base is 2)

• = • •• = • • •

− = − •= • −− − −

>

>

>

>

Double Negative

� The opposite of a negative is

POSITIVE

( ) 1( 1 )x xx− − − − •= =>

1

sum

3 ( 2)

addend addend

2 3

3 ( 2)

3 (

5

2) 1

1

−

−

− −

−

+ =

+ =

+ =

+ −=

−

>

>

>

>


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3. Fractions

3.1. Definitions Fractions Proper fraction

Improper fraction

� Numerator/denominator

o Numerator < Denominator

o Numerator > Denominator

> 2/7

> –7/2

Mixed number � Integer + fraction > –3 ½

Factor � A whole number that divides into another

number

Factors of 18:

1, 2, 3, 6, 9, & 18

Prime Number � A whole number > 1 whose only factors are 1

and itself

> > 2, 3, 5, 7, 11, 13…

Composite Number � A whole number > 1 that is not prime > 4, 6, 8, 9, 10…

Prime Factorization � A composite number written as a product of

prime numbers

218 3 3= • •>

Factor Tree � A method for determining prime factorization

of a number

18

3 6

2 3

Lowest Terms 3 1

or 12 2

� Numerator and denominator have NO

COMMON FACTORS other than 1. Reduce

fractions by cancelling common factors.

� If the numerator and/or denominator has

addition or subtraction, it is not in factored

form. No cancelling of factors

3

4 8NO!!

12 2 6

36 2 6

1

3

4

x y

•= =

• •

+

>

>

Equivalent 1 5

2 10=

� Numbers that represent the same point on a

number line

� Multiplying a number by any fraction equal

to 1 does not change the value of the number

(see Multiplication Identity)

9

9

2

3 272

3 2 27

18

7

x

x

=

= = =

>


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3.2. Least Common Denominators Least Common Multiple (LCM) (Smallest number that

the given numbers will

divide into)

1 METHOD 1 (USE THE LARGEST NUMBER)

1. Start with the largest number. Do the

other numbers divide into it?

2. If yes, you’re done!

3. If no, double the largest number. Do the


4. If yes, you’re done!

5. If no, triple the largest number. Do the


6. Keep going until you’re done

Ex. Find the LCM of 4,6,9

9, no

18, no

27, no

36, YES

☺

2 METHOD 2 (PRIME FACTORIZATION)

1. Determine the prime factorization of

each number

2. The LCM will have every prime factor

that appears in each number. Each

prime factor will appear the number of

times as it appears in the number which

has the most of that factor.


4 6 9

2 2

LCM

2 3 3 3

= 2 2 3

3

3

6=

• • •

3 METHOD 3 (L METHOD)

1. Find a number that divides into at least

two of the numbers

2. Perform the division

3. Repeat steps 1 & 2 until there are no

more numbers that divide into at least

two of the numbers

4. Multiple the leftmost and bottommost

numbers together


L

3

C

4

M

2

=

4

3

2 3

2

6 9

2 1

=

3

2 1 3 36• • • •

Least Common Denominator (LCD)

� The smallest positive number divisible by

all the denominators. The LCD is also the

LCM of the denominators.

1 1 1Ex. , ,

4 6 9

LCD = 36


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3.3. Operations Convert mixed numbers to improper fractions & solve as fractions

Denominator ≠ 0; Always write final answer in lowest terms

Addition/ Subtraction + –

� If the DENOMINATORS are the SAME:

� COMBINE NUMERATORS

� Write answer over common denominator

1 2

4 4

1

4− =

−>

� If the DENOMINATORS are DIFFERENT:

� Write an equivalent expression using the

LEAST COMMON DENOMINATOR

� ADD/SUBTRACT (see “If the

DENOMINATORS are the SAME”)

LCD = 4

1 1 1 2

4 2

1

1 1

4 2

2

42 44

−

=

= − =−

−

>

Multiplication •

� FACTOR numerators and denominators

� Write problem as one big fraction

� CANCEL common factors

� MULTIPLY top × top & bottom× bottom

2 52 25 5

5 6 3

5

35 2

• •• = =

• •

/ /

//>

Division ÷

� RECIPROCATE (flip) the divisor

� MULTIPLY the fractions (See Multiplication) 6

225

25 6

2 256 5 5

25

= =÷ •>

Converting an Improper Fraction to a Mixed Number

1. Numerator ÷ Denominator

2. Whole-number part of the quotient is the

whole-number part of the mixed number.

Remainder

Divisor is the fractional part.

7

2

( 7) 2

3 remainder 1

13

2

=

=

÷

−

= −

−

−

>

Converting a Mixed Number to an Improper Fraction

1. If the mixed number is negative, ignore the sign

in step 2 and add the sign back in step 3

2. (Denominator × whole-number part) +

numerator

Answer to step 23.

Original denominator

13

2

2 1 7

2

3

7

−

+ =

= −

•

>


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4. Decimals

4.1. Operations Addition/ Subtraction + –

� LINE UP the decimals

� “Pad” with 0’s

� Perform operation as though they were whole

numbers

� Remember, the decimal point come straight down

into the answer

1.5

1.92

1.5

.02

.02 .4

.4

+ +

=

+

0

0

>

Multiplication •

� Multiply the decimals as though they were whole

numbers

� Take the results and position the decimal point so the

number of decimal places is equal to the SUM OF THE

NUMBER OF DECIMAL PLACES IN THE ORIGINAL

PROBLEM

1.4 1.5

14 15

2.

.0

1

21

•

= •

=

=

>

Division ÷

� If the DIVISOR contains a DECIMAL POINT

� MOVE THE DECIMAL POINT TO THE RIGHT so that

the divisor is a whole number

� MOVE THE DECIMAL POINT IN THE DIVIDEND THE

SAME NUMBER OF DECIMAL PLACES TO THE RIGHT

� Divide the decimals as though they were whole

numbers

� The decimal point in the answer should BE STRAIGHT

ABOVE THE DECIMAL POINT IN THE DIVIDEND

.02 .25

2 25.2

054

1 01

.5

0

0

12

0

>

To Multiply by Powers of 10 (shortcut)

� Move the DECIMAL POINT TO THE RIGHT the same

number of places as there are zeros in the power of 10

� Move to the right because the number should get

bigger (Add zeros if needed)

67.6 100

67.60

6760.

=

•

=

>

To Divide by Powers of 10 (shortcut)

� Move the DECIMAL POINT TO THE LEFT the same

number of places as there are zeros in the power of 10

� Move to the left because the number should get

smaller (Add zeros if needed)

67.6 1000

067.6

. 6

/

0 76

=

=

>


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5. Conversions

5.1. Percent to Decimal to Fractions To Percent* To Decimal To Fraction

From Percent*

123%

Drop the % sign & divide

by 100. (move the decimal

point 2 digits to the left)

1.23

Write the % value over 100.

Always reduce. 123/100

� If there is a decimal

point, multiply

numerator &

denominator by a power

of 10 to eliminate it.

90.5 90.5 10 905 181

100 100 10 1000 200= = =

�

�

� If there is a fraction part,

write the percent value

as an improper fraction

55

5 35 1 3565 % = 6 100 6 100 600

= =�

From Decimal

1.234

Multiply by 100 (move the

decimal point 2 digits to the

right) & attach the % sign

123.4%

Write number part. Put

decimal part over place

value of right most digit.

Always reduce.

234 1171 1

1000 500=

From Fraction

16

Method 1 - To express as a

mixed number – multiply by

100 50

1 100 1 100

6 1

•• =

36

216 %

3=

Method 2 - To express as a

decimal – convert to

decimal & multiply by 100

.167 • 100 = 16.7%

Perform long division

0.1666

6 1.00006

40364036

4036

4

.167

“%” means “per hundred”


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6. Algebra

6.1. Definitions Constant � A NUMBER > 5

Variable � A LETTER which represents a number > x

Coefficient � A NUMBER associated with variable(s) > 3x

Term � A COMBINATION of coefficients and variable(s), or a

constant

> –3x

Like Terms � Each variable (including the exponent) of the terms is

exactly the same, but they don’t have to be in the same

order

> –3x & 2x > –3x

2 & 2x

2

> –3xy & 2xy

Linear Expression � One or more terms put together by a “+” or “–“

� The variable is to the first power

> –3x + 3

Linear Equation � Has an equals sign

� The variable is to the first power

> –3x+ 3 = 1


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6.2. Operations of Algebra Addition/ Subtraction + – (Combining Like

Terms)

Only COEFFICIENTS of LIKE TERMS are

combined

� Underline terms (or use box &

circle ) as they are combined 2

4 3

3 2 3 5 3

2 3 Can't be combined2 Can't be combined

x x x

x x xy y y

y y

y

x z

y

x

− =

− − + = − +

++

>

>

>

>

Multiplication/ Division

• ÷

COEFFICIENTS AND VARIABLES of ALL

TERMS are combined

2

2 2

2

2 3

2

(4 )( 3 ) 12

( 3 )( 2 )(3) 18

(2 3 ) 6

(2 )( ) 2

)(

x x x

x x x

z z

y y y

y y

y

x x y

y y

x

− = −

− − =

=

=

>

>

>

>


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6.3. Solving Equations with 1 Variable

1111ELIMINATE FRACTIONS

� Multiply both sides of the equation by the LCD

(2 1)0

3

2 1 3

2 1Ex. Solve 0

3

3 3

1 1

0

xx

x

x

x

x

+

+− =

− =

+ −

=

2222REMOVE ANY GROUPING SYMBOLS SUCH AS PARENTHESES

� Use Distributive Property

2 1 3 0x x+ − =

3333SIMPLIFY EACH SIDE

� Combine like terms

2 1 3 0

1 0

x x

x

+ − =− + =

4444GET VARIABLE TERM ON ONE SIDE & CONSTANT TERM ON OTHER SIDE

� Use Addition Property of Equality (moves the WHOLE term)

1 01 1

1

x

x

− −− + =− = −

5555ELIMINATE THE COEFFICIENT OF THE VARIABLE

� Use Multiplication Property of Equality (eliminates PART of a

term)

( ) ( )( )1 11( 1)x

x− = − −

=−

6666CHECK ANSWER ��

� Substitute answer for the variable in the original equation

(1)(1)

0 0

2 10

3

+− =

= √

Note: Occasionally, when solving an equation, the variable “cancels out”:

� If the resulting equation is true (e.g. 5 = 5), then all real numbers are solutions.

� If the resulting equation is false (e.g. 5 = 4), then there are no solutions.


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6.4. Solving for a Specified Variable

1111CIRCLE THE SPECIFIED VARIABLE F

F

C

C C, where T is the temperature is Celsiusand T is the t

5Ex. Solve T =

emperature in

T +329

fFahren

or Theit

2222TREAT THE SPECIFIED VARIABLE AS THE

ONLY VARIABLE IN THE EQUATION & USE THE

STEPS FOR SOLVING LINEAR EQUATIONS

1. Eliminate fractions

2. Remove parenthesis

3. Simplify each side

4. Get variable term on one side & constant

term on other side (use

addition/subtraction)

5. Eliminate the coefficient of the variable

(use multiplication/division)

6. Check answer �

( )

( )

F C

F C

F C

-32 -32

9 9

5 5

5T = T +32

9

5T -32 = T

9

9T -32 = T

5


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6.5. Expressions vs. Equations Expressions Equations

Definition One or more terms put together by

a “+” or “–“ 2Ex. 2 1x x+ +

Expression=Expression

2Ex. 2 1 0x x+ + =

Equivalent Evaluate both for x = 2 & get

same answer 2

2

Ex. 2 1

3 1

x x

x x x

+ +

= + − +

2

2

Ex. 2 1 0

2 4 2 0

Same solution

2x x

x x

=

+

×+ +

+ =

Expand Opposite of factoring –

rewrite without parenthesis

Ex. 2( 1)

2 2

x

x

+

= +

Often used in solving equations

Factor Opposite of expanding –

rewrite as a product of smaller

expressions

Ex. 2 2

2( 1)

x

x

+

= +


Cancel Only within a fraction in

factored form

2 2Ex. 3

2

2 3

( )x

x

++

= + +


Simplify/ Evaluate/ Add/Subtract/ Multiply/Divide An equalivent expression with

a smaller number of parts

Cancelling common factors (top

& bottom) & collecting like terms

2

2 2Ex. 3

4

2 6 8

2 2 2

( )x

x x

+/+

+ += + =

~denominators stay when adding

fractions


~denominators are eliminated

when simplifying equations

Evaluate for a number Substitute the given number

(put it in parenthesis) &

simplify

2

2

base is 2, n

Ex. Evaluate

f

ot 2

or 2

= ( 2) ( 2)

( 2)

!

4

x x = −

− • −

− −

=

−

Used to check answers

(

Ex. Evaluate

0 2 for 2

0 2

0 0

so 2 is a solut

2)

ion

x x

−= + = −= +

=−

√

Solve Find all possible values of the

variable(s) Ex. 2 0

2

x

x

+ =

= −


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7. Word Problems

7.1. Vocabulary Addition � The sum of a and b

� The total of a, b, and c

� 8 more than a

� a increased by 3

a + b

a + b + c

a + 8

a + 3

Subtraction � a subtracted from b

� The difference of a and b

� 8 less than a

� a decreased by 3

b – a

a – b

a – 8

a – 3

Multiplication � 1/2 of a

� The product of a and b

� twice a

� a times 3

(1/2)a

a b•

2a

3a

Division � The quotient of a and b

� 8 into a

� a divided by 3

a b÷

a/8

a/3

General � Variable words

� Multiplication words

� Equals words

what, how much, a number

of

is, was, would be

Percent Word Problems

� 50% of 60 is what

� 50% of what is 30

� What % of 60 is 30

50% • 60 = a

50% • a = 30

(a%) • 60 = 30


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7.2. Formulas Commission � Commission = commission rate • sales amount

Sales Tax Total Price

� Sales tax = sales tax rate • purchase price

� Total price = purchase price + Sales tax

Amount of Discount Sale Price

� Amount of discount = discount rate • original price

� Sale price = original price – Amount of discount

Simple Interest � Simple interest = principal • interest rate • time

Percent increase/decrease � Percent increase/decrease =

amount of increase/decrease

amount

original

• 100


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7.3. Steps for Solving

1111 UNDERSTAND THE PROBLEM

� As you use information, cross it out or

underline it.

� Remember units!

Kevin’s age is 3 years more than twice Jane’s age.

The sum of their ages is 39. How old are Kevin

and Jane?

2222 NAME WHAT x IS

� Start your LET statement

� x can only be one thing

� When in doubt, choose the smaller thing

Let x = Jane’s age (years)

3333 DEFINE EVERYTHING ELSE IN TERMS OF x 2x + 3 = Kevin’s age

4444 WRITE THE EQUATION Kevin's age + Jane's

2

age 39=

33 9x x+ =+

5555 SOLVE THE EQUATION 3 3 3

( 3

9x + =

− ) 3 3x+ +1

39

3 1

3 312

( 3)

36

1 1

x

x

=

=

=

−+

6666 ANSWER THE QUESTION

� Answer must include units!

Jane’s age = 12 years

Kevin’s age = 2(12) + 3

= 27 years

7777 CHECK 12 27 39+ = √


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7.4. Ratios & Proportions Rate Ratio

� The quotient of two quantities

� Used to compare different kinds of rates 1 person 2 gallons

, 100 people 50 miles

Proportion � A statement that TWO RATIOS or RATES are

EQUAL

� On a map 50 miles is represented by 25 inches.

10 miles would be represented by how many

inches?

10 (miles) 50 (miles)

(inches) 25 (inches)x=

Cross Product (shortcut)

� Method for solving for x in a proportion

� Multiply diagonally across a proportion

� If the cross products are equal, the proportion is

true. If the cross products are not equal, the

proportion is false

10 50

25x=

10 25 50

250 50

250

550

x

x

x

x

• = •=

=

=


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7.5. Geometry Terminology Perimeter/

Circumference � Measures the length around the outside of the figure (RIM);

the answer is in the same units as the sides

Area � Measures the size of the enclosed region of the figure; the

answer is in square units

Surface Area � Measures the outside area of a 3 dimensional figure; the

answer is in square units

Volume � Measures the enclosed region of a 3 dimensional figure; the

answer is in cubic units

Formulas Square s

� PERIMETER: P = 4s

� AREA: A = s2

(s = side)

Rectangle l w

� PERIMETER: P = 2l + 2w

� AREA: A = lw

(l = length, w = width)

Parallelogram

� DEFINITION: a four-sided figure with two pairs of parallel

sides.

� PERIMETER: P = 2h + 2b

� AREA: A = hb

(h = height, b = base)

Trapezoid

� DEFINITION: a four-sided figure with one pair of parallel

sides

� PERIMETER: P = b1 + b2 + other two sides

� AREA: A = ((b1 + b2) / 2)h

(h = height, b = base)

Rectangular Solid

� SURFACE AREA: A =2hw + 2lw + 2lh

� VOLUME: V = lwh

(l = length, w = width, h = height)

Circle d

r

� CIRCUMFERENCE: C = 2π r

� AREA: A = π r2

( )1 = radius, = diameter, 2

r d r d=

Sphere

� SURFACE AREA: A = 2dπ

� VOLUME: V = 34

3rπ

Triangle a

c

h

b

� PERIMETER: P = a + b + c

� AREA: 1

2A bh=

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